Rational Numbers Class 8 GSEB Notes

This GSEB Class 8 Maths Notes Chapter 1 Rational Numbers covers all the important topics and concepts as mentioned in the chapter.

Rational Numbers Class 8 GSEB Notes

Natural numbers:

• The counting numbers are known as natural numbers. 1, 2, 3, 4, … are natural numbers. The collection of all-natural numbers is denoted by ‘N’
  N = {1, 2, 3, 4,…}
• The smallest natural number is 1.
• There are infinite natural numbers.

Whole numbers:

• If ‘0’ is included in the collection of natural numbers, then the collection of numbers 0, 1, 2, 3, … is known as whole numbers. It is denoted by ‘W’
  W = {0, 1, 2, 3, 4,…}
• The smallest whole number is 0.
• There is no greatest whole number.

Rational numbers:

• A number of the form $\frac{p}{q}$, where p and q are integers and q ≠ 0 is called a rational number.
  $\frac{0}{5}, \frac{3}{7}, \frac{(-5)}{9}, 19, \frac{7}{(-13)}$, etc. are all rational numbers.
• $\frac{0}{5}$ is a rational number but $\frac{5}{0}$ is not a rational number. (∵ q = 0)
• 0 is a whole number but not a natural number. Every natural number is a whole \number but every whole number is not a natural number.
• Zero is a rational number because we can divide zero by a non-zero number.
• There are infinite rational numbers between two rational numbers.

Basic operation :

• Addition
• Subtraction
• Multiplication
• Division

Properties of addition :

• Closure property
• Commutative property
• Associative property
• Property of zero OR additive identity
• Property of additive inverse

1. Closure property: The sum of two rational numbers is a rational number. If x and y are two rational numbers, then (x + y) is also a rational number.

2. Commutative property: Addition of rational numbers is commutative. If x and y are two rational numbers, then x + y = y + x.

3. Associative property: The addition of any three rational numbers is associative. If x, y and z are any three rational numbers, then (x + y) + z = x + (y + z).

4. Additive identity: The sum of a rational number and zero (0) is the same rational number. If x is a rational number, then x + 0 = 0 + x = x.

Property of additive inverse:

• If we add two same rational numbers having opposite signs, the sum is zero.
• If x is rational number, then
  x + (- x) = (- x) + x = 0
• The negative of x is denoted by (- x) and vice versa.

Properties of subtraction:

• Closure property: If x and y are two rational numbers then x – y is a rational number.
• Commutative property: Commutative property does not hold for subtraction of rational numbers,
  x – y ≠ y  – x
• Associative property: The subtraction of rational numbers is not associative. If x. y and z are any three rational numbers, then (x- y) – z ≠ x- (y – z).

Properties of multiplication:

• Closure property
• Commutative property
• Associative property
• Multiplicative identity
• Distributive property of multiplication (over addition)

1. Closure property: If x and y are two rational numbers, then x × y is also a rational number.

2. Commutative property: For any two rational numbers x and y,
x × y = y × x

3. Associative property: If x, y and z are any three rational numbers, then
(x × y) × z = x × (y × z)

4. Multiplicative identity: If x is any rational number, then
x × 1 = 1 × x = x
∴ 1 is called multiplicative identity.

5. Distributive property of multiplication (over addition) : If x, y and z are any three rational numbers, then
x × (y + z) = x × y + x × z

Properties of division:
If x and y are any two rational numbers and y ≠ 0, then x ÷ y is always a rational number.
For any rational number x
x ÷ 1 = x and x ÷ (- 1) = (-x)

For every non-zero rational number
x ÷ x = 1
x ÷ (-x) = (- 1)
(-x) – x = (- 1)

Property of multiplicative inverse:
If $\frac{p}{q}$ is a rational number (p, q ≠ 0), then $\frac{q}{p}$ is the multiplicative inverse (reciprocal) of $\frac{p}{q}$.

Their product is always 1.
$\frac{p}{q} \times \frac{q}{p}$ = 1

Note:

• We can also represent a rational number on a number line. Let’s say the rational number Is If you want to plot It accurately on the number line, divide the number line between two whole numbers between which $\frac{x}{y}$ lies Into y equal parts and plot It on the xth part between those two numbers.
  e.g.. To represent on a number line, make five equal parts between 0 and 1.
• Between two rational numbers x and y, there Is a rational number $\frac{x+y}{2}$
• We can find as many rational numbers between x and y as we want.